3.104 \(\int x^2 (a+b \log (c x^n))^2 \log (d (e+f x^2)^m) \, dx\)

Optimal. Leaf size=630 \[ \frac{2 b (-e)^{3/2} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{2 b (-e)^{3/2} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{2 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 b^2 (-e)^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{(-e)^{3/2} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac{(-e)^{3/2} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{16 a b e m n x}{9 f}-\frac{16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{52 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3 \]

[Out]

(-16*a*b*e*m*n*x)/(9*f) + (52*b^2*e*m*n^2*x)/(27*f) - (4*b^2*m*n^2*x^3)/27 - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt
[f]*x)/Sqrt[e]])/(27*f^(3/2)) - (16*b^2*e*m*n*x*Log[c*x^n])/(9*f) + (8*b*m*n*x^3*(a + b*Log[c*x^n]))/27 + (4*b
*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) + (2*e*m*x*(a + b*Log[c*x^n])^2)/(3*f
) - (2*m*x^3*(a + b*Log[c*x^n])^2)/9 - ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^
(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2*b^2*n^2*x^3*Log[d*
(e + f*x^2)^m])/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(
e + f*x^2)^m])/3 + (2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (
2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (((2*I)/9)*b^2*e^(3/2)*m
*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (((2*I)/9)*b^2*e^(3/2)*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqr
t[e]])/f^(3/2) - (2*b^2*(-e)^(3/2)*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (2*b^2*(-e)^(3/2)*
m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.06533, antiderivative size = 630, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 17, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.607, Rules used = {2305, 2304, 2378, 302, 205, 2351, 2295, 2324, 12, 4848, 2391, 2353, 2296, 2330, 2317, 2374, 6589} \[ \frac{2 b (-e)^{3/2} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{2 b (-e)^{3/2} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{2 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 b^2 (-e)^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{(-e)^{3/2} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac{(-e)^{3/2} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{16 a b e m n x}{9 f}-\frac{16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{52 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m],x]

[Out]

(-16*a*b*e*m*n*x)/(9*f) + (52*b^2*e*m*n^2*x)/(27*f) - (4*b^2*m*n^2*x^3)/27 - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt
[f]*x)/Sqrt[e]])/(27*f^(3/2)) - (16*b^2*e*m*n*x*Log[c*x^n])/(9*f) + (8*b*m*n*x^3*(a + b*Log[c*x^n]))/27 + (4*b
*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) + (2*e*m*x*(a + b*Log[c*x^n])^2)/(3*f
) - (2*m*x^3*(a + b*Log[c*x^n])^2)/9 - ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^
(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2*b^2*n^2*x^3*Log[d*
(e + f*x^2)^m])/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(
e + f*x^2)^m])/3 + (2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (
2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (((2*I)/9)*b^2*e^(3/2)*m
*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (((2*I)/9)*b^2*e^(3/2)*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqr
t[e]])/f^(3/2) - (2*b^2*(-e)^(3/2)*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (2*b^2*(-e)^(3/2)*
m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2))

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac{2 b^2 n^2 x^4}{27 \left (e+f x^2\right )}-\frac{2 b n x^4 \left (a+b \log \left (c x^n\right )\right )}{9 \left (e+f x^2\right )}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 \left (e+f x^2\right )}\right ) \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 f m) \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+\frac{1}{9} (4 b f m n) \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx-\frac{1}{27} \left (4 b^2 f m n^2\right ) \int \frac{x^4}{e+f x^2} \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 f m) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 \left (e+f x^2\right )}\right ) \, dx+\frac{1}{9} (4 b f m n) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{f^2 \left (e+f x^2\right )}\right ) \, dx-\frac{1}{27} \left (4 b^2 f m n^2\right ) \int \left (-\frac{e}{f^2}+\frac{x^2}{f}+\frac{e^2}{f^2 \left (e+f x^2\right )}\right ) \, dx\\ &=\frac{4 b^2 e m n^2 x}{27 f}-\frac{4}{81} b^2 m n^2 x^3+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 m) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac{(2 e m) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}-\frac{\left (2 e^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{3 f}+\frac{1}{9} (4 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{(4 b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}+\frac{\left (4 b e^2 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x^2} \, dx}{9 f}-\frac{\left (4 b^2 e^2 m n^2\right ) \int \frac{1}{e+f x^2} \, dx}{27 f}\\ &=-\frac{4 a b e m n x}{9 f}+\frac{4 b^2 e m n^2 x}{27 f}-\frac{8}{81} b^2 m n^2 x^3-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{\left (2 e^2 m\right ) \int \left (\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}-\sqrt{f} x\right )}+\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}+\sqrt{f} x\right )}\right ) \, dx}{3 f}+\frac{1}{9} (4 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{(4 b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}-\frac{\left (4 b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{9 f}-\frac{\left (4 b^2 e^2 m n^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} \sqrt{f} x} \, dx}{9 f}\\ &=-\frac{16 a b e m n x}{9 f}+\frac{16 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}-\frac{4 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left ((-e)^{3/2} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}-\sqrt{f} x} \, dx}{3 f}+\frac{\left ((-e)^{3/2} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}+\sqrt{f} x} \, dx}{3 f}-\frac{\left (4 b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{3 f}-\frac{\left (4 b^2 e^{3/2} m n^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=-\frac{16 a b e m n x}{9 f}+\frac{52 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}-\frac{16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left (2 b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac{\left (2 b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac{\left (2 i b^2 e^{3/2} m n^2\right ) \int \frac{\log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}+\frac{\left (2 i b^2 e^{3/2} m n^2\right ) \int \frac{\log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=-\frac{16 a b e m n x}{9 f}+\frac{52 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}-\frac{16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 i b^2 e^{3/2} m n^2 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 i b^2 e^{3/2} m n^2 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=-\frac{16 a b e m n x}{9 f}+\frac{52 b^2 e m n^2 x}{27 f}-\frac{4}{27} b^2 m n^2 x^3-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}-\frac{16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac{8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 i b^2 e^{3/2} m n^2 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 i b^2 e^{3/2} m n^2 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 b^2 (-e)^{3/2} m n^2 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.43001, size = 1128, normalized size = 1.79 \[ \frac{-4 b^2 f^{3/2} m n^2 x^3-6 b^2 f^{3/2} m \log ^2\left (c x^n\right ) x^3-6 a^2 f^{3/2} m x^3+8 a b f^{3/2} m n x^3-12 a b f^{3/2} m \log \left (c x^n\right ) x^3+8 b^2 f^{3/2} m n \log \left (c x^n\right ) x^3+2 b^2 f^{3/2} n^2 \log \left (d \left (f x^2+e\right )^m\right ) x^3+9 b^2 f^{3/2} \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3+9 a^2 f^{3/2} \log \left (d \left (f x^2+e\right )^m\right ) x^3-6 a b f^{3/2} n \log \left (d \left (f x^2+e\right )^m\right ) x^3+18 a b f^{3/2} \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3-6 b^2 f^{3/2} n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3+52 b^2 e \sqrt{f} m n^2 x+18 b^2 e \sqrt{f} m \log ^2\left (c x^n\right ) x+18 a^2 e \sqrt{f} m x-48 a b e \sqrt{f} m n x+36 a b e \sqrt{f} m \log \left (c x^n\right ) x-48 b^2 e \sqrt{f} m n \log \left (c x^n\right ) x-18 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2(x)-18 b^2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2\left (c x^n\right )-4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-18 a^2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+12 a b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-12 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x)+36 a b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x)-36 a b e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )+12 b^2 e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )+36 b^2 e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) \log \left (c x^n\right )+9 i b^2 e^{3/2} m n^2 \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+6 i b^2 e^{3/2} m n^2 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-18 i a b e^{3/2} m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-18 i b^2 e^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-9 i b^2 e^{3/2} m n^2 \log ^2(x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )-6 i b^2 e^{3/2} m n^2 \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+18 i a b e^{3/2} m n \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+18 i b^2 e^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+6 i b e^{3/2} m n \left (3 a-b n+3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+6 i b e^{3/2} m n \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-18 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+18 i b^2 e^{3/2} m n^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m],x]

[Out]

(18*a^2*e*Sqrt[f]*m*x - 48*a*b*e*Sqrt[f]*m*n*x + 52*b^2*e*Sqrt[f]*m*n^2*x - 6*a^2*f^(3/2)*m*x^3 + 8*a*b*f^(3/2
)*m*n*x^3 - 4*b^2*f^(3/2)*m*n^2*x^3 - 18*a^2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b*e^(3/2)*m*n*ArcTan
[(Sqrt[f]*x)/Sqrt[e]] - 4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 36*a*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x
)/Sqrt[e]]*Log[x] - 12*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 18*b^2*e^(3/2)*m*n^2*ArcTan[(Sqr
t[f]*x)/Sqrt[e]]*Log[x]^2 + 36*a*b*e*Sqrt[f]*m*x*Log[c*x^n] - 48*b^2*e*Sqrt[f]*m*n*x*Log[c*x^n] - 12*a*b*f^(3/
2)*m*x^3*Log[c*x^n] + 8*b^2*f^(3/2)*m*n*x^3*Log[c*x^n] - 36*a*b*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^
n] + 12*b^2*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 36*b^2*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]
]*Log[x]*Log[c*x^n] + 18*b^2*e*Sqrt[f]*m*x*Log[c*x^n]^2 - 6*b^2*f^(3/2)*m*x^3*Log[c*x^n]^2 - 18*b^2*e^(3/2)*m*
ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - (18*I)*a*b*e^(3/2)*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I
)*b^2*e^(3/2)*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^2*e^(3/2)*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f
]*x)/Sqrt[e]] - (18*I)*b^2*e^(3/2)*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*a*b*e^(3/2)*m
*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^2*e^(3/2)*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (9*
I)*b^2*e^(3/2)*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^2*e^(3/2)*m*n*Log[x]*Log[c*x^n]*Log[1
+ (I*Sqrt[f]*x)/Sqrt[e]] + 9*a^2*f^(3/2)*x^3*Log[d*(e + f*x^2)^m] - 6*a*b*f^(3/2)*n*x^3*Log[d*(e + f*x^2)^m] +
2*b^2*f^(3/2)*n^2*x^3*Log[d*(e + f*x^2)^m] + 18*a*b*f^(3/2)*x^3*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 6*b^2*f^(3/
2)*n*x^3*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 9*b^2*f^(3/2)*x^3*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + (6*I)*b*e^(3/
2)*m*n*(3*a - b*n + 3*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b*e^(3/2)*m*n*(-3*a + b*n - 3
*b*Log[c*x^n])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^2*e^(3/2)*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e
]] + (18*I)*b^2*e^(3/2)*m*n^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2))

________________________________________________________________________________________

Maple [F]  time = 13.078, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m),x)

[Out]

int(x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m),x, algorithm="fricas")

[Out]

integral((b^2*x^2*log(c*x^n)^2 + 2*a*b*x^2*log(c*x^n) + a^2*x^2)*log((f*x^2 + e)^m*d), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))**2*ln(d*(f*x**2+e)**m),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*x^2*log((f*x^2 + e)^m*d), x)